Local lower norm estimates for dyadic maximal operators and related Bellman functions

TL;DR

This paper establishes lower bounds for localized dyadic maximal operators in various norm settings, extending results to general homogeneous tree-like structures in probability spaces.

Contribution

It introduces new lower norm estimates for dyadic maximal operators in both Euclidean and abstract probabilistic contexts, generalizing previous bounds.

Findings

Lower $L^q$ bounds for dyadic maximal operators

Weak $L^p$ bounds for localized operators

Extension to homogeneous tree-like structures

Abstract

We provide lower $L^q$ and weak $L^p$-bounds for the localized dyadic maximal operator on $R^n$, when the local $L^1$ and the local $L^p$ norm of the function are given. We actually do that in the more general context of homo- geneous tree-like families in probability spaces.